Lecture 13 Linear quadratic Lyapunov theory · PDF fileEE363 Winter 2008-09 Lecture 13 Linear quadratic Lyapunov theory • the Lyapunov equation • Lyapunov stability conditions

EE363 Winter 2008-09

Lecture 13

Linear quadratic Lyapunov theory

• the Lyapunov equation

• Lyapunov stability conditions

• the Lyapunov operator and integral

• evaluating quadratic integrals

• analysis of ARE

• discrete-time results

• linearization theorem

13–1

The Lyapunov equation

the Lyapunov equation is

ATP + PA + Q = 0

where A, P, Q ∈ Rn×n, and P, Q are symmetric

interpretation: for linear system x = Ax, if V (z) = zTPz, then

V (z) = (Az)TPz + zTP (Az) = −zTQz

i.e., if zTPz is the (generalized)energy, then zTQz is the associated(generalized) dissipation

linear-quadratic Lyapunov theory: linear dynamics, quadratic Lyapunovfunction

Linear quadratic Lyapunov theory 13–2

we consider system x = Ax, with λ1, . . . , λn the eigenvalues of A

if P > 0, then

• the sublevel sets are ellipsoids (and bounded)

• V (z) = zTPz = 0 ⇔ z = 0

boundedness condition: if P > 0, Q ≥ 0 then

• all trajectories of x = Ax are bounded(this means ℜλi ≤ 0, and if ℜλi = 0, then λi corresponds to a Jordanblock of size one)

• the ellipsoids {z | zTPz ≤ a} are invariant


Stability condition

if P > 0, Q > 0 then the system x = Ax is (globally asymptotically)stable, i.e., ℜλi < 0

to see this, note that

V (z) = −zTQz ≤ −λmin(Q)zTz ≤ −λmin(Q)

λmax(P )zTPz = −αV (z)

where α = λmin(Q)/λmax(P ) > 0


An extension based on observability

(Lasalle’s theorem for linear dynamics, quadratic function)

if P > 0, Q ≥ 0, and (Q,A) observable, then the system x = Ax is(globally asymptotically) stable

to see this, we first note that all eigenvalues satisfy ℜλi ≤ 0

now suppose that v 6= 0, Av = λv, ℜλ = 0

then Av = λv = −λv, so

∥

∥

∥Q1/2v

∥

∥

∥

2

= v∗Qv = −v∗(

ATP + PA)

v = λv∗Pv − λv∗Pv = 0

which implies Q1/2v = 0, so Qv = 0, contradicting observability (by PBH)

interpretation: observability condition means no trajectory can stay in the“zero dissipation” set {z | zTQz = 0}


An instability condition

if Q ≥ 0 and P 6≥ 0, then A is not stable

to see this, note that V ≤ 0, so V (x(t)) ≤ V (x(0))

since P 6≥ 0, there is a w with V (w) < 0; trajectory starting at w does notconverge to zero

in this case, the sublevel sets {z | V (z) ≤ 0} (which are unbounded) areinvariant


The Lyapunov operator

the Lyapunov operator is given by

L(P ) = ATP + PA

special case of Sylvester operator

L is nonsingular if and only if A and −A share no common eigenvalues,i.e., A does not have pair of eigenvalues which are negatives of each other

• if A is stable, Lyapunov operator is nonsingular

• if A has imaginary (nonzero, iω-axis) eigenvalue, then Lyapunovoperator is singular

thus if A is stable, for any Q there is exactly one solution P of Lyapunovequation ATP + PA + Q = 0


Solving the Lyapunov equation

ATP + PA + Q = 0

we are given A and Q and want to find P

if Lyapunov equation is solved as a set of n(n + 1)/2 equations inn(n + 1)/2 variables, cost is O(n6) operations

fast methods, that exploit the special structure of the linear equations, cansolve Lyapunov equation with cost O(n3)

based on first reducing A to Schur or upper Hessenberg form


The Lyapunov integral

if A is stable there is an explicit formula for solution of Lyapunov equation:

P =

∫

∞

0

etATQetA dt

to see this, we note that

ATP + PA =

∫

∞

0

(

ATetATQetA + etAT

QetAA)

dt

=

∫

∞

0

(

d

dtetAT

QetA

)

dt

= etATQetA

∣

∣

∣

∞

0

= −Q


Interpretation as cost-to-go

if A is stable, and P is (unique) solution of ATP + PA + Q = 0, then

V (z) = zTPz

= zT

(∫

∞

0

etATQetA dt

)

z

=

∫

∞

0

x(t)TQx(t) dt

where x = Ax, x(0) = z

thus V (z) is cost-to-go from point z (with no input) and integralquadratic cost function with matrix Q


if A is stable and Q > 0, then for each t, etATQetA > 0, so

P =

∫

∞

0

etATQetA dt > 0

meaning: if A is stable,

• we can choose any positive definite quadratic form zTQz as thedissipation, i.e., −V = zTQz

• then solve a set of linear equations to find the (unique) quadratic formV (z) = zTPz

• V will be positive definite, so it is a Lyapunov function that proves A isstable

in particular: a linear system is stable if and only if there is a quadratic

Lyapunov function that proves it


generalization: if A stable, Q ≥ 0, and (Q, A) observable, then P > 0

to see this, the Lyapunov integral shows P ≥ 0

if Pz = 0, then

0 = zTPz = zT

(∫

∞

0

etATQetA dt

)

z =

∫

∞

0

∥

∥

∥Q1/2etAz

∥

∥

∥

2

dt

so we conclude Q1/2etAz = 0 for all t ≥ 0

this implies that Qz = 0, QAz = 0, . . . , QAn−1z = 0, contradicting(Q,A) observable


Monotonicity of Lyapunov operator inverse

suppose ATPi + PiA + Qi = 0, i = 1, 2

if Q1 ≥ Q2, then for all t, etATQ1e

tA ≥ etATQ2e

tA

if A is stable, we have

P1 =

∫

∞

0

etATQ1e

tA dt ≥

∫

∞

0

etATQ2e

tA dt = P2

in other words: if A is stable then

Q1 ≥ Q2 =⇒ L−1(Q1) ≥ L−1(Q2)

i.e., inverse Lyapunov operator is monotonic, or preserves matrixinequality, when A is stable

(question: is L monotonic?)


Evaluating quadratic integrals

suppose x = Ax is stable, and define

J =

∫

∞

0

x(t)TQx(t) dt

to find J , we solve Lyapunov equation ATP + PA + Q = 0 for P

then, J = x(0)TPx(0)

in other words: we can evaluate quadratic integral exactly, by solving a setof linear equations, without even computing a matrix exponential


Controllability and observability Grammians

for A stable, the controllability Grammian of (A, B) is defined as

Wc =

∫

∞

0

etABBTetATdt

and the observability Grammian of (C, A) is

Wo =

∫

∞

0

etATCTCetA dt

these Grammians can be computed by solving the Lyapunov equations

AWc + WcAT + BBT = 0, ATWo + WoA + CTC = 0

we always have Wc ≥ 0, Wo ≥ 0;Wc > 0 if and only if (A, B) is controllable, andWo > 0 if and only if (C, A) is observable


Evaluating a state feedback gain

consider

x = Ax + Bu, y = Cx, u = Kx, x(0) = x0

with closed-loop system x = (A + BK)x stable

to evaluate the quadratic integral performance measures

Ju =

∫

∞

0

u(t)Tu(t) dt, Jy =

∫

∞

0

y(t)Ty(t) dt

we solve Lyapunov equations

(A + BK)TPu + Pu(A + BK) + KTK = 0

(A + BK)TPy + Py(A + BK) + CTC = 0

then we have Ju = xT0 Pux0, Jy = xT

0 Pyx0


Lyapunov analysis of ARE

write ARE (with Q ≥ 0, R > 0)

ATP + PA + Q − PBR−1BTP = 0

as(A + BK)TP + P (A + BK) + (Q + KTRK) = 0

with K = −R−1BTP

we conclude: if A + BK stable, then P ≥ 0 (since Q + KTRK ≥ 0)

i.e., any stabilizing solution of ARE is PSD

if also (Q,A) is observable, then we conclude P > 0

to see this, we need to show that (Q + KTRK, A + BK) is observable

if not, there is v 6= 0 s.t.

(A + BK)v = λv, (Q + KTRK)v = 0


which implies

v∗(Q + KTRK)v = v∗Qv + v∗KTRKv = ‖Q1/2v‖2 + ‖R1/2Kv‖2 = 0

so Qv = 0, Kv = 0

(A + BK)v = Av = λv, Qv = 0

which contradicts (Q,A) observable

the same argument shows that if P > 0 and (Q, A) is observable, thenA + BK is stable


Monotonic norm convergence

suppose that A + AT < 0, i.e., (symmetric part of) A is negative definite

can express as ATP + PA + Q = 0, with P = I, Q > 0

meaning: xTPx = ‖x‖2 decreases along every nonzero trajectory, i.e.,

• ‖x(t)‖ is always decreasing monotonically to 0

• x(t) is always moving towards origin

this implies A is stable, but the converse is false: for a stable system, weneed not have A + AT < 0

(for a stable system with A + AT 6< 0, ‖x(t)‖ converges to zero, but notmonotonically)


for a stable system we can always change coordinates so we havemonotonic norm convergence

let P denote the solution of ATP + PA + I = 0

take T = P−1/2

in new coordinates A becomes A = T−1AT ,

A + AT = P 1/2AP−1/2 + P−1/2ATP 1/2

= P−1/2(

PA + ATP)

P−1/2

= −P−1 < 0

in new coordinates, convergence is obvious because ‖x(t)‖ is alwaysdecreasing


Discrete-time results

all linear quadratic Lyapunov results have discrete-time counterparts

the discrete-time Lyapunov equation is

ATPA − P + Q = 0

meaning: if xt+1 = Axt and V (z) = zTPz, then ∆V (z) = −zTQz

• if P > 0 and Q > 0, then A is (discrete-time) stable (i.e., |λi| < 1)

• if P > 0 and Q ≥ 0, then all trajectories are bounded(i.e., |λi| ≤ 1; |λi| = 1 only for 1 × 1 Jordan blocks)

• if P > 0, Q ≥ 0, and (Q, A) observable, then A is stable

• if P 6> 0 and Q ≥ 0, then A is not stable


Discrete-time Lyapunov operator

the discrete-time Lyapunov operator is given by L(P ) = ATPA − P

L is nonsingular if and only if, for all i, j, λiλj 6= 1(roughly speaking, if and only if A and A−1 share no eigenvalues)

if A is stable, then L is nonsingular; in fact

P =

∞∑

t=0

(AT )tQAt

is the unique solution of Lyapunov equation ATPA − P + Q = 0

the discrete-time Lyapunov equation can be solved quickly (i.e., O(n3))and can be used to evaluate infinite sums of quadratic functions, etc.


Converse theorems

suppose xt+1 = Axt is stable, ATPA − P + Q = 0

• if Q > 0 then P > 0

• if Q ≥ 0 and (Q,A) observable, then P > 0

in particular, a discrete-time linear system is stable if and only if there is aquadratic Lyapunov function that proves it


Monotonic norm convergence

suppose ATPA − P + Q = 0, with P = I and Q > 0

this means ATA < I, i.e., ‖A‖ < 1

meaning: ‖xt‖ decreases on every nonzero trajectory; indeed,‖xt+1‖ ≤ ‖A‖‖xt‖ < ‖xt‖

when ‖A‖ < 1,

• stability is obvious, since ‖xt‖ ≤ ‖A‖t‖x0‖

• system is called contractive since norm is reduced at each step

the converse is false: system can be stable without ‖A‖ < 1


now suppose A is stable, and let P satisfy ATPA − P + I = 0

take T = P−1/2

in new coordinates A becomes A = T−1AT , so

AT A = P−1/2ATPAP−1/2

= P−1/2(P − I)P−1/2

= I − P−1 < I

i.e., ‖A‖ < 1

so for a stable system, we can change coordinates so the system iscontractive


Lyapunov’s linearization theorem

we consider nonlinear time-invariant system x = f(x), where f : Rn → Rn

suppose xe is an equilibrium point, i.e., f(xe) = 0, and letA = Df(xe) ∈ Rn×n

the linearized system, near xe, is ˙δx = Aδx

linearization theorem:

• if the linearized system is stable, i.e., ℜλi(A) < 0 for i = 1, . . . , n, thenthe nonlinear system is locally asymptotically stable

• if for some i, ℜλi(A) > 0, then the nonlinear system is not locallyasymptotically stable


stability of the linearized system determines the local stability of thenonlinear system, except when all eigenvalues are in the closed lefthalfplane, and at least one is on the imaginary axis

examples like x = x3 (which is not LAS) and x = −x3 (which is LAS)show the theorem cannot, in general, be tightened

examples:

eigenvalues of Df(xe) conclusion about x = f(x)−3, − 0.1 ± 4i, − 0.2 ± i LAS near xe

−3, − 0.1 ± 4i, 0.2 ± i not LAS near xe

−3, − 0.1 ± 4i, ± i no conclusion


Proof of linearization theorem

let’s assume xe = 0, and express the nonlinear differential equation as

x = Ax + g(x)

where ‖g(x)‖ ≤ K‖x‖2

suppose that A is stable, and let P be unique solution of Lyapunovequation

ATP + PA + I = 0

the Lyapunov function V (z) = zTPz proves stability of the linearizedsystem; we’ll use it to prove local asymptotic stability of the nonlinearsystem


V (z) = 2zTP (Az + g(z))

= zT (ATP + PA)z + 2zTPg(z)

= −zTz + 2zTPg(z)

≤ −‖z‖2 + 2‖z‖‖P‖‖g(z)‖

≤ −‖z‖2 + 2K‖P‖‖z‖3

= −‖z‖2(1 − 2K‖P‖‖z‖)

so for ‖z‖ ≤ 1/(4K‖P‖),

V (z) ≤ −1

2‖z‖2 ≤ −

1

2λmax(P )zTPz = −

1

2‖P‖zTPz


finally, using

‖z‖2 ≤1

λmin(P )zTPz

we have

V (z) ≤λmin(P )

16K2‖P‖2=⇒ ‖z‖ ≤

1

4K‖P‖=⇒ V (z) ≤ −

1

2‖P‖V (z)

and we’re done

comments:

• proof actually constructs an ellipsoid inside basin of attraction ofxe = 0, and a bound on exponential rate of convergence

• choice of Q = I was arbitrary; can get better estimates using other Qs,better bounds on g, tighter bounding arguments . . .


Integral quadratic performance

consider x = f(x), x(0) = x0

we are interested in the integral quadratic performance measure

J(x0) =

∫

∞

0

x(t)TQx(t) dt

for any fixed x0 we can find this (approximately) by simulation andnumerical integration

(we’ll assume the integral exists; we do not require Q ≥ 0)


Lyapunov bounds on integral quadratic performance

suppose there is a function V : Rn → R such that

• V (z) ≥ 0 for all z

• V (z) ≤ −zTQz for all z

then we have J(x0) ≤ V (x0), i.e., the Lyapunov function V serves as anupper bound on the integral quadratic cost

(since Q need not be PSD, we might not have V ≤ 0; so we cannotconclude that trajectories are bounded)


to show this, we note that

V (x(T )) − V (x(0)) =

∫ T

0

V (x(t)) dt ≤ −

∫ T

0

x(t)TQx(t) dt

and so

∫ T

0

x(t)TQx(t) dt ≤ V (x(0)) − V (x(T )) ≤ V (x(0))

since this holds for arbitrary T , we conclude

∫

∞

0

x(t)TQx(t) dt ≤ V (x(0))


Integral quadratic performance for linear systems

for a stable linear system, with Q ≥ 0, the Lyapunov bound is sharp, i.e.,there exists a V such that

• V (z) ≥ 0 for all z

• V (z) ≤ −zTQz for all z

and for which V (x0) = J(x0) for all x0

(take V (z) = zTPz, where P is solution of ATP + PA + Q = 0)


Lecture 13 Linear quadratic Lyapunov theory · PDF fileEE363 Winter 2008-09 Lecture 13 Linear quadratic Lyapunov theory • the Lyapunov equation • Lyapunov stability conditions

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